Tuesday, February 2, 2010

Homework #1

Here is the first homework:

1. Let M be a finitely generated R-module and P in Spec(R). Then M_p = 0 if and only if P does not contain the annihilator of M.

2. Suppose R is a domain. Then one can view R_m as a subset of the fraction field of R, for every maximal ideal m. Show that R is the intersection of all R_m where m ranges over all maximal ideals.

3. Let R be a ring, and S,T two m.c. subsets of R. Let U be the image of T in S^(-1)R. Show that (ST)^(-1)R = U^(-1)(S^(-1)R).

4. Show that if S is a m.c. subset of R, then S^(-1)R is isomorphic to R[{x_u}]/(ux_u - 1).

5. Let R be a ring and let M be an R-module. Suppose that {f_i} is a set of elements of R that generate the unit ideal. Prove that
a) If m \in M goes to 0 in each M_f_i, then m = 0, and
b) If m_i \in M_f_i are elements such that m_i and m_j go to the same element of M_(f_if_j), then there is an element m \in M such that m goes to m_i in M_f_i.

6. A m.c. subset S of R is called saturated if for all a,b in R, ab in S implies a or b is in S. Show that S is saturated if and only if the complement of S is a union of prime ideals.

7. Let S_0 be the set of nonzerodivisors of a ring R. Show that S_0 is a saturated m.c. subset. The ring S_0^(-1)R is called the total ring of fractions of R.
i) Show that S_0 is the largest m.c. subset for which R --> S_0^(-1)R is injective.
ii) Show that every element of S_0^(-1)R is either a zerodivisor or a unit.
iii) Show that every ring in which every non-unit is a zerodivisor is equal to its total ring of fractions.

Please email me if you have any questions, or if you see a typo in the questions.

Forgot to include the due date for the assignment: Feb 11th.

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